Chapter 10 - Introduction to Differential Equations - 10.1 Solving Differential Equations - Exercises - Page 505: 34

$$y^3=-\frac{3}{2}(x^{-2}-1)$$

By separation of variables, we have $$y^2dy=x^{-3}dx$$ then by integration, we get $$\frac{1}{3} y^3=-\frac{1}{2}x^{-2}+c\Longrightarrow y^3=-\frac{3}{2}x^{-2}+A.$$ Now, since $y(1)=0$, then $A= \frac{3}{2}$. So the general solution is given by $$y^3=-\frac{3}{2}(x^{-2}-1)$$